OFFSET
1,3
COMMENTS
For n>2, each row begins with 0 and ends with n+1. If the term in row n and column k is denoted by U(n,k), then U(n,n-2)=A105163(n-1).
Row sums: A000225 (-1+2^n)
Alternating row sums: A137470
Limiting row: 0,2,5,13,34,89,..., even-indexed Fibonacci numbers
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+x,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First six rows:
1
1...2
1...1...3
1...1...3...4
1...1...2...8...5
1...1...2...6...17...6
First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + x;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A210876 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210877 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 30 2012
STATUS
approved